Local Whittle likelihood estimators and tests for spatial lattice data

- Local Whittle estimators and associated test statistics are established.
- The asymptotics of these statistics are elucidated.
- Different asymptotic schemes in analyzing spatial data are considered, PID and MID.
- An application of the proposed method for the Mercer and Hall data is presented.

Likelihood methods for large spatial datasets on a d-dimensional lattice are often difficult, if not infeasible, to implement due to the computational limitation. When the number of observations gets increasingly large, it is challenging to evaluate the exact likelihood even if normality is assumed, since the covariance matrix is huge. To alleviate the computational difficulties, one can approximate the exact parametric likelihood through the Whittle likelihood, which is based on periodogram ordinates and spectral density functions. By applying fast Fourier transform, it requires less computational operations to calculate these magnitudes. In this paper, we define a local Whittle likelihood function, which can be used to estimate spectral density functions of general spatial lattice processes and a local Whittle likelihood ratio statistic for testing particular parametric spatial classes. When the bandwidth used is large, this method amounts to ordinary Whittle likelihood, while for moderate or small bandwidth, the method is essentially nonparametric estimation of spectral density functions. In addition, the asymptotic properties of the proposed estimator and associated test statistic are derived. It is shown that the asymptotic distribution of the test statistic is not affected by the underlying spatial asymptotic framework or the shape of the sampling regions. Simulation results and real data examples are also reported to illustrate the finite sample performance of the methodology.